Most constitutional monarchies got that way due to incremental change generally caused by political crises. Switching from a monarchy to a republic usually done as a response to one of these crises; no crisis usually means the monarch keeps the crown.

You also have an issue of what to replace the monarch with. Most constitutional monarchies have parliamentary systems of government where the legislature has supremacy. However, you still need a supreme executive to run a government when the legislature fails. The process of picking that person is very politically important and had inherent risks to it. For some countries, keeping the monarch as the on/off switch is easier than dealing with the headache of choosing a President.